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We have obtained approximate bound state solutions of Schr ödinger wave equation with modified quadratic Yukawa plus q-deformed Eckart potential Using Parametric Nikiforov-Uvarov (NU) method. However, we obtained numerical energy eigenvalues and un-normalized wave function using confluent hypergeometric function (Jacobi polynomial). With some modifications, our potential reduces to a well-known potential such as Poschl-Teller and exponential inversely quadratic potential. Numerical bound state energies were carried out using a well-designed Matlab algorithm while the plots were obtained using origin software. The result obtained is in agreement with that of the existing literature.

Researchers have put on their interest over the years with the aim of investigating the bound state solutions of relativistic and nonrelativistic wave equations for different potentials. A few of these potentials have been solved exactly [

The Eckart potential is one of the most important exponential-type potentials in the literature and it has been widely used in physics and chemical physics [

The Yukawa potential, which is also known as the screened Coulomb potential, is greatly important with applications cutting across nuclear physics and condensed matter physics [

With approximate analytical solution of the Yukawa potential with arbitrary momenta using the Nikiforov-Uvarov method, they obtained approximate analytical solutions of the radial Schrödinger equation for the Yukawa potential, and the energy eigenvalues and the corresponding eigen functions are calculated in closed forms [

V ( r ) = v o e − 2 α r r 2 + v 1 ( 1 + e − 2 α r 1 − q e − 2 α r ) + v 2 ( 4 e − 2 α r ( 1 − q e − 2 α r ) 2 ) , (1)

where v o , v 1 , v 2 are the potential depth, q is the deformation parameter, r is the inter-nuclear distance and α is the screening parameter. It can be deduced that when v o = v 1 = 0 , v 2 = − v 1 and q = − 1 , the above combined potential reduces to the Poschl-Teller potential. When v 1 = v 2 = 0 , the potential reduces to Exponential inversely quadratic potential. Using the parametric NU method, we derive the energy bound state solutions and their wave functions of the Schrodinger equation for the modified quadratic Yukawa plus q-deformed Eckart potential, analytically and numerically. The rest of the manuscript is organized as follows: In Section 2, a review of Nikiforov-Uvarov method is presented. The radial solutions of Schrodinger equation are presented in Section 3. The numerical solutions are presented in Section 4 while discussion and conclusion are presented in Sections 5 and 6 respectively.

The parametric form is simply using parameters to obtain explicitly energy eigenvalues and it is based on the solutions of a generalized second order linear differential equation with special orthogonal functions [

Given a second order differential equation of the form

ψ ″ ( s ) + τ ¯ σ ( s ) ψ ′ ( s ) + σ ¯ ( s ) σ 2 ( s ) ψ ( s ) = 0 (2)

where σ and σ ¯ are polynomials at most degree and τ ¯ is first degree polynomials. The parametric generalization of the N-U method is given by the generalized hypergeometric-type equation

ψ ″ ( s ) + c 1 − c 2 s s ( 1 − c 3 s ) ψ ′ + 1 s 2 ( 1 − c 3 s ) 2 [ − χ 1 s 2 + χ 2 s − χ 3 ] ψ ( s ) = 0 (3)

Thus Equation (2) can be solved by comparing it with Equation (3) and the following polynomials are obtained

τ ¯ ( s ) = ( c 1 − c 2 s ) , σ ( s ) = s ( 1 − c 3 s ) , σ ¯ ( s ) = − χ 1 s 2 + χ 2 s − χ 3

The parameters obtainable from Equation (3) serve as an important tool for finding the energy eigenvalue and eigenfunctions.

Where c 1 , c 2 and c 3 are parametric constants, other parametric constants are given below

{ c 4 = 1 2 ( 1 − c 1 ) c 5 = 1 2 ( c 2 − 2 c 3 ) c 6 = c 5 2 + χ 1 c 7 = 2 c 4 c 5 − χ 2 c 8 = c 4 2 + χ 3 c 9 = c 3 c 7 + c 3 2 c 8 + c 6 c 10 = c 1 + 2 c 4 + 2 c 8 c 11 = c 2 − 2 c 5 + 2 [ c 9 + c 3 c 8 ] c 12 = c 4 + c 8 c 13 = c 5 − [ c 9 + c 3 c 8 ] (4)

The energy equation is given as

c 2 n − ( 2 n + 1 ) c 5 + ( 2 n + 1 ) [ c 9 + c 3 c 8 ] + n ( n + 1 ) c 3 + c 7 + 2 c 3 c 8 + 2 c 8 c 9 = 0 (5)

The total wave function is given by

ψ ( s ) = ϕ ( s ) χ n ( s ) = N n s c 12 ( 1 − c 3 s ) c 13 P [ c 10 , c 11 ] ( 1 − 2 c 3 s ) (6)

The Schrödinger equation is given by

d 2 R d r 2 + [ 2 μ ℏ 2 [ E n l − V ( r ) ] − l ( l + 1 ) r 2 ] R ( r ) = 0 (7)

where μ is the reduced mass, E n l is the rotational vibrational energy spectra of the diatomic molecules, ℏ is the reduced Planck’s constant, n and l are the radial and orbital angular momentum quantum numbers, respectively [

Substituting Equation (1) into (7) gives

d 2 R d r 2 + [ 2 μ ℏ 2 [ E − v o e − 2 α r r 2 − v 1 ( 1 + e − 2 α r 1 − q e − 2 α r ) − v 2 ( 4 e − 2 α r ( 1 − q e − 2 α r ) 2 ) ] − l ( l + 1 ) r 2 ] R ( r ) = 0 (8)

Let’s define suitable approximation to the centrifugal term as

1 r 2 = 4 α 2 e − 2 α r ( 1 − q e − 2 α r ) 2 (9)

Substituting Equation (9) into (8) gives

d 2 R d r 2 + { 2 μ ℏ 2 [ E − 4 α 2 v o e − 4 α r ( 1 − q e − 2 α r ) 2 − v 1 ( 1 + e − 2 α r 1 − q e − 2 α r ) − v 2 ( 4 e − 2 α r ( 1 − q e − 2 α r ) 2 ) ] − 4 α 2 e − 2 α r ( 1 − q e − 2 α r ) 2 l ( l + 1 ) } R ( r ) = 0 (10)

In order to present Equation (10) in a form that is conformable to Equation (3) then, there is a need for the transformation

s = q e − 2 α r = d 2 R d r 2 = 4 α 2 q 2 e − 4 α r d 2 R d s 2 + 4 α 2 q e − 2 α r d R d s and e − 2 α r = s q (11)

Substituting (11) into (10) and simplifying gives

d 2 R d s 2 + q ( 1 − s ) q s ( 1 − s ) d R d s + 1 s 2 q 2 ( 1 − s ) 2 [ − ( ε 2 q 2 − A 2 + B 2 q ) s 2 + ( 2 ε 2 q 2 + B 2 q 2 − B 2 q − C 2 q − q λ ) s − ( ε 2 q 2 + B 2 q 2 ) ] R ( s ) = 0 (12)

where

μ E 2 α 2 ℏ 2 = − ε 2 , 2 μ v o ℏ 2 = A 2 , μ v 1 2 α 2 ℏ 2 = B 2 , μ v 2 α 2 ℏ 2 = C 2 and l ( l + 1 ) = λ (13)

Comparing with parametric NU

{ χ 1 = ε 2 q 2 − A 2 + B 2 q , χ 2 = 2 ε 2 q 2 + B 2 q 2 − B 2 q − C 2 q − q λ χ 3 = ε 2 q 2 + B 2 q 2 , c 1 = c 2 = c 3 = 1 (14)

{ c 4 = 1 2 ( 1 − c 1 ) = 0 c 5 = 1 2 ( c 2 − 2 c 3 ) = − 1 2 c 6 = c 5 2 + χ 1 = 1 4 + ε 2 q 2 − A 2 + B 2 q c 7 = 2 c 4 c 5 − χ 2 = − 2 ε 2 q 2 − B 2 q 2 + B 2 q + C 2 q + q λ c 8 = c 4 2 + χ 3 = ε 2 q 2 + B 2 q 2 c 9 = c 3 c 7 + c 3 2 c 8 + c 6 = 2 B 2 q + C 2 q + q λ + 1 4 − A 2 c 10 = c 1 + 2 c 4 + 2 c 8 = 1 + 2 ε 2 q 2 + B 2 q 2 = 1 + 2 ε q + 2 B 2 q 2 c 11 = c 2 − 2 c 5 + 2 [ c 9 + c 3 c 8 ] = 2 + 2 [ 2 B 2 q + C 2 q + q λ + 1 4 − A 2 + ε 2 q 2 + B 2 q 2 ] c 12 = c 4 + c 8 = ε 2 q 2 + B 2 q 2 c 13 = c 5 − [ c 9 + c 3 c 8 ] = − 1 2 − 2 B 2 q + C 2 q + q λ + 1 4 − A 2 − ε 2 q 2 + B 2 q 2 (15)

The energy can be calculated by substituting Equation (14) and Equation (15) into (5) with simple mathematical algebraic simplification. Hence, the energy eigen equation become

n − ( 2 n + 1 ) ( − 1 2 ) + ( 2 n + 1 ) [ 2 B 2 q + C 2 q + q λ + 1 4 − A 2 + ε 2 q 2 + B 2 q 2 ] + n 2 − n + ( − 2 ε 2 q 2 − B 2 q 2 + B 2 q + C 2 q + q λ ) + 2 ε 2 q 2 + 2 B 2 q 2 + 2 ( ε 2 q 2 + B 2 q 2 ) ( 2 B 2 q + C 2 q + q λ + 1 4 − A 2 ) = 0 (16)

Factorizing and rearranging Equation (16)

ε = − [ n 2 + n + 1 2 + ( 2 n + 1 ) 2 B 2 q + C 2 q + q λ + 1 4 − A 2 + ( 2 n + 1 ) B 2 q 2 + B 2 q + C 2 q + q λ + B 2 q 2 + B 2 q 2 ( 2 B 2 q + C 2 q + q λ + 1 4 − A 2 ) ] q ( 2 n + 1 ) + 2 q 2 ( 2 B 2 q + C 2 q + q λ + 1 4 − A 2 ) (17)

E n l = − 2 α 2 ℏ 2 μ [ n 2 + n + 1 2 + ( 2 n + 1 ) 2 ( μ v 1 2 α 2 h 2 ) q + ( μ v 2 α 2 h 2 ) q + q l ( l + 1 ) + 1 4 − ( 2 μ v o h 2 ) + ( 2 n + 1 ) ( μ v 1 2 α 2 h 2 ) q 2 + ( μ v 1 2 α 2 ℏ 2 ) q + ( μ v 2 α 2 ℏ 2 ) q + q l ( l + 1 ) + ( μ v 1 2 α 2 ℏ 2 ) q 2 + ( μ v 1 2 α 2 ℏ 2 ) q 2 ( 2 ( μ v 1 2 α 2 ℏ 2 ) q + ( μ v 2 α 2 ℏ 2 ) q + q l ( l + 1 ) + 1 4 − ( 2 μ v o ℏ 2 ) ) q ( 2 n + 1 ) + 2 q 2 ( 2 ( μ v 1 2 α 2 ℏ 2 ) q + ( μ v 2 α 2 ℏ 2 ) q + q l ( l + 1 ) + 1 4 − ( 2 μ v o ℏ 2 ) ) ] 2 (18)

Case 1:

If we replace parameter v o = v 1 = 0 , v 2 = − v 1 and q = − 1 , then the potential function reduces to Poschl-Teller potential which is written as:

V ( r ) = − 4 v 1 e − 2 α r ( 1 + e − 2 α r ) 2 (19)

The energy equation for Poschl-Teller potential becomes:

E n l = − 2 α 2 ℏ 2 μ [ n 2 + n + 1 2 + ( 2 n + 1 ) ( μ v 1 α 2 ℏ 2 ) − l ( l + 1 ) + 1 4 + ( μ v 1 α 2 ℏ 2 ) − l ( l + 1 ) ( 2 n + 1 ) + 2 ( μ v 1 α 2 ℏ 2 ) − l ( l + 1 ) + 1 4 ] 2 (20)

Case 2:

If we also replace parameter v 1 = v 2 = 0 we obtain exponential inversely quadratic potential which is expressed as

V ( r ) = v 0 e − 2 α r r 2 (21)

Also, the energy equation becomes:

E n l = − 2 α 2 ℏ 2 μ [ n 2 + n + 1 2 + ( 2 n + 1 ) q l ( l + 1 ) + 1 4 − ( 2 μ v o h 2 ) + q l ( l + 1 ) q ( 2 n + 1 ) + 2 q 2 ( + q l ( l + 1 ) + 1 4 − ( 2 μ v o ℏ 2 ) ) ] 2 (22)

The wave function for the proposed potential can be calculated using equation (6), expressed in terms of Jacobi polynomial and substituting the necessary parameters. Thus,

ψ ( s ) = Φ ( s ) x n ( s ) = N n s − μ E 2 α 2 ℏ 2 q 2 + μ v 1 2 α 2 ℏ 2 q 2 ( 1 − s ) [ − 1 2 − 2 μ v 1 2 α 2 ℏ 2 q + μ v 2 α 2 ℏ 2 q + q l ( l + 1 ) + 1 4 − 2 μ v o ℏ 2 − − μ E 2 α 2 ℏ 2 q 2 + μ v 1 2 α 2 ℏ 2 q 2 ] × P [ 1 + 2 − μ E 2 α 2 ℏ 2 q + 2 μ v 1 2 α 2 ℏ 2 q 2 , 2 + 2 [ 2 μ v 1 2 α 2 ℏ 2 q + μ v 2 α 2 ℏ 2 q + q l ( l + 1 ) + 1 4 − 2 μ v o ℏ 2 + − μ E 2 α 2 ℏ 2 q 2 + μ v 1 2 α 2 ℏ 2 q 2 ] ] ( 1 − 2 s ) (21)

where N n is the normalization constant.

The numerical computation was carried out using MATLAB 8.0 software while graphical plot was carried out using origin software. The computation parameters have been incorporated in the numerical table. The energy equation for the proposed potential is given in Equation (18). The computation was carried for various range of values of the screening i.e. for large and small values of α .

The results of the numerical computation that were carried out for the screening parameter α = (0.1 to 0.5) with the potential depths V_{0} = 0.3, V_{1} = 0.1, V_{2} = 0.2. Tables 1-5 show that for α = (0.1 to 0.5), the numerical bound state energy decreases with an increase in quantum state. It is also observed that the energy increases with a decrease in screening parameter α. However, the energy spectral diagram as shown in Figures 1-5 shows unique quantization of different energy levels with respect to the quantum state. The numerical bound state energies obtained for the various values of α are predominantly negative which the sufficient and necessary condition for bound state solutions. Also the negative bound state eigenvalues obtained show that the potential is suitable for describing particle state and their bound state properties.

We used the Nikiforov-Uvarov method to compute bound state solution of non-relativistic wave equation. We developed a new potential model which is the

n | E n l ( α = 0.1 ) , l = 0 | E n l ( α = 0.1 ) , l = 1 | E n l ( α = 0.1 ) , l = 2 | E n l ( α = 0.1 ) , l = 3 |
---|---|---|---|---|

0 | −0.358723860422 | −0.373451796993 | −0.402514181567 | −0.445267610477 |

1 | −0.475369012878 | −0.491714416155 | −0.523889432048 | −0.571048112966 |

2 | −0.600647379426 | −0.618682417516 | −0.654093357336 | −0.705803142808 |

3 | −0.734864206154 | −0.754635641966 | −0.793364191204 | −0.849724805080 |

4 | −0.878257368475 | −0.899795273869 | −0.941895045046 | −1.002973867370 |

5 | −1.031005151520 | −1.054328978440 | −1.099835318390 | −1.165677990550 |

6 | −1.193240706270 | −1.218363132520 | −1.267299510740 | −1.337937137990 |

7 | −1.365064290600 | −1.391993618710 | −1.444375701660 | −1.519829600630 |

8 | −1.546552317100 | −1.575293981950 | −1.631132241580 | −1.711417064680 |

n | E n l ( α = 0.2 ) , l = 0 | E n l ( α = 0.2 ) , l = 1 | E n l ( α = 0.2 ) , l = 2 | E n l ( α = 0.2 ) , l = 3 |
---|---|---|---|---|

0 | −0.433780559429 | −0.494363866152 | −0.610559494829 | −0.776706656632 |

1 | −0.681533081008 | −0.755505128052 | −0.894725310230 | −1.089765618020 |

2 | −0.965541546776 | −1.053407891020 | −1.216334999480 | −1.440874970160 |

3 | −1.287362985910 | −1.389313601350 | −1.576247051710 | −1.830596392810 |

4 | −1.647812113560 | −1.763917890310 | −1.974991302070 | −2.259310793850 |

5 | −2.047347843110 | −2.177631633140 | −2.412903173260 | −2.727276909810 |

6 | −2.486246379460 | −2.630710863480 | −2.890202843360 | −3.234674069180 |

7 | −2.964683306910 | −3.123322720630 | −3.407039979430 | −3.781629551740 |

8 | −3.482775365680 | −3.655580683110 | −3.963519540590 | −4.368235808860 |

n | E n l ( α = 0.3 ) , l = 0 | E n l ( α = 0.3 ) , l = 1 | E n l ( α = 0.3 ) , l = 2 | E n l ( α = 0.3 ) , l = 3 |
---|---|---|---|---|

0 | −0.523052525651 | −0.661370123967 | −0.919655127988 | −1.28258956659 |

1 | −0.913253413467 | −1.097274964830 | −1.423624712520 | −1.86403452990 |

2 | −1.388776992580 | −1.619048677850 | −2.014441440220 | −2.53317685538 |

3 | −1.952153981160 | −2.228660567050 | −2.693363583100 | −3.29078021582 |

4 | −2.604372373890 | −2.927007369790 | −3.461069410510 | −4.13731637556 |

5 | −3.345895349130 | −3.714551928370 | −4.317951065080 | −5.07308614428 |

6 | −4.176968473750 | −4.591556423970 | −5.264249578690 | −6.09828854546 |

7 | −5.097733958340 | −5.558180233760 | −6.300120794220 | −7.21305986496 |

8 | −6.108279821150 | −6.614525733030 | −7.425669769950 | −8.41749630564 |

n | E n l ( α = 0.4 ) , l = 0 | E n l ( α = 0.4 ) , l = 1 | E n l ( α = 0.4 ) , l = 2 | E n l ( α = 0.4 ) , l = 3 |
---|---|---|---|---|

0 | −0.627591541101 | −0.874606577279 | −1.32739264725 | −1.95796715215 |

1 | −1.167338531860 | −1.522463516870 | −2.11456444317 | −2.89392389718 |

2 | −1.865350921070 | −2.326614313680 | −3.05852731571 | −3.98749113984 |

3 | −2.722717879280 | −3.289031117730 | −4.16067450669 | −5.23951982221 |

4 | −3.739774576790 | −4.410499607300 | −5.42170064184 | −6.65050810572 |

5 | −4.916658364750 | −5.691393534650 | −6.84198775364 | −8.22076276659 |

6 | −6.253435948480 | −7.131913188680 | −8.42176256885 | −9.95048181780 |

7 | −7.750143631940 | −8.732175506250 | −10.1611679045 | −11.8397983528 |

8 | −9.406802865400 | −10.49225332580 | −12.0602981088 | −13.8888049619 |

n | E n l ( α = 0.5 ) , l = 0 | E n l ( α = 0.5 ) , l = 1 | E n l ( α = 0.5 ) , l = 2 | E n l ( α = 0.5 ) , l = 3 |
---|---|---|---|---|

0 | −0.749317899709 | −1.13386318743 | −1.83234253525 | −2.80079527677 |

1 | −1.439799855380 | −2.03415440794 | −2.96841426538 | −4.17870059279 |

2 | −2.388456731380 | −3.18228093061 | −4.35154186289 | −5.80426205934 |

3 | −3.589627738370 | −4.57947155574 | −5.98306986743 | −7.67835319798 |

4 | −5.041897269490 | −6.22617321021 | −7.86363513976 | −9.80146692921 |

5 | −6.744744343520 | −8.12258737350 | −9.99357686299 | −12.1738997806 |

6 | −8.697932398990 | −10.2688182127 | −12.3730919053 | −14.7958400397 |

7 | −10.90133853290 | −12.6649249983 | −15.0023022605 | −17.6674127310 |

8 | −13.35489256650 | −15.3109439381 | −17.8812875111 | −20.7887040137 |

modified quadratic Yukawa plus q-deformed Eckart potential. This potential reduces to a well-known potential called the Poschl-Teller potential and the exponential inversely quadratic potential. We also implement some algorithms to carry out numerical computation for the resulting energy equations for various values of screening parameters. Our analytic and computational results for the proposed potential are in agreement with the existing literature.

The authors are very grateful to the reviewer for the significant suggestions which have led to the improvement of this article.

The authors have no conflict of interest as regarding the publication of this article.

Antia, A.D., Okon, I.B., Akankpo, A.O. and Usanga, J.B. (2020) Non-Relativistic Bound State Solutions of Modified Quadratic Yukawa plus q-Deformed Eckart Potential. Journal of Applied Mathematics and Physics, 8, 660-671. https://doi.org/10.4236/jamp.2020.84051